A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that |xy| ≤ |x||y|. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use (c-star-algebras) ...

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115 views

Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
5
votes
0answers
116 views

invariant subspace of a Hardy space

Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
4
votes
0answers
107 views

Invertibility of elements in a Banach algebra

Let $X=L^1\cap L^2$, and $\hat{X}$ be the Banach algebra of the image under Fourier transform of $X$. Then do the unital extension $1\dot{+}\hat{X}$ of $X$ by adding a constant function with the norm ...
4
votes
0answers
82 views

Open map in Banach algebra

I'm having trouble showing a certian function is open and can be extended. Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded ...
4
votes
0answers
108 views

Biduals generated by projections

This question is motivated by a similar question recently posed at MO: http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras In this setting, let $B$ be a Banach algebra ...
3
votes
0answers
47 views

Do inclusions of Banach algebras preserve spectral radius?

Let $f : A_1 \to A_2$ be an injective homomorphism of Banach algebras. It's a standard fact that if $f$ is has closed range, i.e. $A_1$ is embedded as a closed subalgebra of $A_2$, then for every $x ...
3
votes
0answers
116 views

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
3
votes
0answers
67 views

In relation with the set of Fredholm perturbation elements

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
3
votes
0answers
129 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
3
votes
0answers
61 views

Let $K$ be a circle. Describe the spectra of two subalgebras of $C(K)$

Suppose $K=\{\lambda\in\mathbb{C}: 1<\vert\lambda\vert<2\};$ put $f(\lambda)=\lambda$. Let $A$ be the smallest closed subalgebra of $C(K$) that contains $1$ and $f$. Let $B$ be the smallest ...
3
votes
0answers
101 views

Density of operators

I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by ...
3
votes
0answers
96 views

Does the non-emptiness of the spectrum of an element of a Banach algebra depend on the Axiom of Choice?

One of the most basic results in functional analysis states that the spectrum of any element of a Banach algebra is non-empty. The proof, as most people might have seen, makes use of Liouville's ...
2
votes
0answers
19 views

Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$ in the Banach algebra $A(\mathbb R)$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
2
votes
0answers
29 views

Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
2
votes
0answers
46 views

How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
2
votes
0answers
109 views

Functional analysis problem involving maximal ideal space

For the past week, I've been trying to solve, as a practice homework exercise, Problem 6 of Chapter 11 in Rudin's Functional Analysis, but have not gotten very far it seems. The problem is as follows: ...
2
votes
0answers
51 views

Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
2
votes
0answers
62 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
2
votes
0answers
43 views

Character space of $L^{1} (\mathbb Z)$

I have a question about the Gelfand and norm topologies on the character space of $L^{1} (\mathbb Z)$. Are the Gelfand and norm topologies equal, on the character space of $L^{1} (\mathbb ...
2
votes
0answers
59 views

The group algebra is separable

Let $G$ be a locally compact Hausdorff group. Is the group algebra $L^1(G)$ separable? Would you please help me or introduce references that can help me. Thanks
2
votes
0answers
35 views

Initial topology of the spectrum mapping $\sigma$

Let $\mathcal{A}$ be a Banach algebra, the map $\sigma$ maps each element $a\in\mathcal{A}$ to its spectrum $\sigma(a)$, which is a compact subset of $\mathbb{C}$. The collection of compact subsets ...
2
votes
0answers
51 views

Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?

Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
1
vote
0answers
10 views

When do Fourier series and Fourier transform coincide

The other day I proved that if $f \in \ell^1 (\mathbb Z)$ then its Gelfand transform $\widehat{f}$ is a map $S^1 \to S^1$ such that $$ \widehat{f}(z) = \sum_{k \in \mathbb Z}f(k) z^k$$ and that ...
1
vote
0answers
40 views

Need help understanding this proof

I need help understanding the following: If $A$ is a (complex) banach algebra and $I$ is a proper modular ideal then $\overline{I}$ is also proper. Proof. Let $u\in A$ be such that $a-ua, a-au \in ...
1
vote
0answers
32 views

What are these spectra (part 1)

I needed some insight into spectra in Banach algebras. To this end I tried to calculate a few examples. I will post them not all in one post. Could somebody please tell me if they are correct? Here ...
1
vote
0answers
22 views

Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
1
vote
0answers
37 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
1
vote
0answers
29 views

Why the space of Fourier-Stieltjes transforms on a compact groups is same as the space of FT on compact groups?

Let $G$ be a locally compact group and we put $L^{1}(G)= \ \text {The space of all complex functions which are integrable with respect to the Haar measure of } \ G.$ Let $\Gamma$ is the dual group ...
1
vote
0answers
36 views

Limit of the spectrum in Banach algebra

Let $A$ an unital complex Banach algebra, $a_{n} $ is a sequence such that $\lim_{n\to \infty}a_{n}=a$. What is the relation between $\lim_{n\to \infty}\sigma(a_{n})$ and $\sigma(a)$. I think that it ...
1
vote
0answers
40 views

ultrapowers of a matrix

1)Let A, B, C, D be a Banach algebras, and U be a free ultrafilter. Can we see that ultrapowers of \begin{pmatrix} A & B \\ C & D \end{pmatrix} equal to \begin{pmatrix} (A)_U & ...
1
vote
0answers
45 views

Integration of rational function on Banach algebra

I do not follow the proof of this Theorem Theorem Suppose$R(\lambda) = P(\lambda) + \sum_{m,k}c_{m,k}(\lambda - \alpha_m)^{-k}$ is a rational function with poles at the points $\alpha_m$. ($P$ ...
1
vote
0answers
71 views

Representation of subspaces as complemented subspaces

Let $X$ be any separable Banach space. The Banach-Mazur theorem states (astonishingly) that $X$ is isometric to a closed subspace of $C(\Delta)$, the space of continuous functions on the cantor set ...
1
vote
0answers
15 views

Closure of the set of fredholm perturbation

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
1
vote
0answers
29 views

Definite positive measure and GNS representation

Let $G$ be a locally compact group. Let $\mu$ be a positive definite complex measure ([D, p295]): we have $\mu(f*f^*)\geq 0$ for any compact support continuons function $f \in C_c(G)$. In [D, p ...
1
vote
0answers
67 views

Commutative Noetherian Banach algebra.

Prove that: 1) Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers. 2) Every commutative real unital Noetherian Banach algebra ...
1
vote
0answers
80 views

Approximation of certain continuous functions by analytic functions

Let $f\in C(S^{1},M_{n}(\mathbb{C}))$ be a unitary. Does there exist an analytic unitary function $g$ from $S^{1}$ to $M_{n}(\mathbb{C})$ that approximates $f$?
1
vote
0answers
46 views

Resolution of Identity

I am reading the twelfth chapter of Rudin's Functional Analysis and am unable to understand two comments he makes in passing in the section on Resolution of Identity. Let E be a resolution of identity ...
1
vote
0answers
81 views

Semisimple commutative Banach algebra

On a semisimple commutative Banach algebra all Banach algebra norms are equivalent. Is this true without assuming semisimplicity?
1
vote
0answers
51 views

$\ast$-homomorphism

Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
1
vote
0answers
74 views

Unitary equivalent

In general, if two irreducible representations of a $C^*$-algebra have the same kernel we can say this two representations are approximately unitarily equivalent. When our $C^*$-algebra is GCR, how to ...
1
vote
0answers
153 views

Proving properties of exponential map on a Banach algebra

$$\exp(a) := \sum\frac {a^k}{k!}$$ Can you help me prove that: $\exp$ is well defined (i.e. converges for all $a$ in $A$) $\exp$ is continuous $\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
1
vote
0answers
44 views

The hermitian element $h=\sum_{n=1}^\infty \frac{p_{n}}{3^{n}}$ generates $C_{0}(\Omega)$‎

‎Please help me to solve the following problem‎ : Let $\Omega$ be a locally compact Hausdorff space‎, ‎and suppose that the $C^{*}$-algebra $C_{0}(\Omega)$ is generated by a sequence of projections ...
1
vote
0answers
64 views

When is the orbit of a vector a minimal sequence? When does an operator have a minimal orbit vector?

Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors. For ...
0
votes
0answers
7 views

How to prove the set of fourier multipliers is a banach algebra?

Hi I am new here at math stack Exchange, this is my first question, hope you guys can help me out:) Suppose $F\colon L^2(\mathbb{R} ) \to L^2(\mathbb{R})$ is the Fourier transform given by ...
0
votes
0answers
14 views

Characters of $\ell^p(\mathbb Z)$

The other day I showed that the character space $\Omega (\ell^1 (\mathbb Z))$ is homeomorphic to $S^1$. Now I am wondering if there are similar identifications for $\ell^p(\mathbb Z)$ when $p \in ...
0
votes
0answers
36 views

How to prove $\Omega (A)$ is weak star closed

If $A$ is a unital complex commutative Banach algebra to show that the Gelfand spectrum $\Omega (A)$ is weak star closed how to finish the following arguemnt: My idea was to consider $\tau_n \in ...
0
votes
0answers
30 views

Need some help finishing this proof about characters in Banach algebras

I tried to prove: Let $A$ be a commutative unital complex Banach algebra. Then there is a bijection between the maximal ideals in $A$ and the set of non-zero homomorphisms $A \to \mathbb C$. But I ...
0
votes
0answers
51 views

Is it a typo in the statement of this theorem

Consider the following theorem: If $I$ is a modular maximal ideal of a unital abelian algebra $A$, then $A/I$ is a field. It is a basic fact of algebra that if $R$ is a commutative unital ring then ...
0
votes
0answers
8 views

Spectrum in infinite hilbert space

Show that if a complex hilbert space $H$ is separable, then for every compact set $K$, there exists a bounded linear operator $T:H→H$ such that $σ(T)=K$S
0
votes
0answers
10 views

How to use $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in Banach sapace $C([0, T];M^{p,1})$?

(For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ...